Definite Integration Question - None

Question

Paragraph for Question nos. 599 and 600
Let $f(x)$ be a polynomial of degree 3 such that $f(0)=1$, $f(1)=2$ and zero is a critical point of $f(x)$ having no local extreme.

If the value of definite integral $\displaystyle\int_{-1}^{1} \frac{f(x)}{\sqrt{x^2+7}}\, dx$ is equal to $2\ln\!\left(\dfrac{\sqrt{a}+1}{\sqrt{b}+c}\right)$, then the value of $(a+b+c)$ is:

Accepted Answer

Step 1: Determine f(x)

Let f(x) = ax³ + bx² + cx + d

From f(0) = 1: d = 1

From f'(x) = 3ax² + 2bx + c, and f'(0) = 0: c = 0

From f''(x) = 6ax + 2b, and f''(0) = 0 (inflection point): b = 0

So f(x) = ax³ + 1

From f(1) = 2: a + 1 = 2, so a = 1

Therefore: f(x) = x³ + 1

Step 2: Evaluate the integral

I = ∫₋₁¹ (x³ + 1)/√(x² + 7) dx

Split: I = ∫₋₁¹ x³/√(x² + 7) dx + ∫₋₁¹ 1/√(x² + 7) dx

First integral: x³/√(x² + 7) is odd, so it equals 0

Second integral: ∫₋₁¹ 1/√(x² + 7) dx = [sinh⁻¹(x/√7)]₋₁¹ = ln(x + √(x² + 7))|₋₁¹

= ln(1 + √8) - ln(-1 + √8)

= ln(1 + 2√2) - ln(2√2 - 1)

= ln[(1 + 2√2)/(2√2 - 1)]

Step 3: Rationalize and match form

Multiply by (2√2 + 1)/(2√2 + 1):

= ln[(1 + 2√2)(2√2 + 1)/(7)] = ln[(2√2 + 1 + 8 + 2√2)/7] = ln[(9 + 4√2)/7]

= 2ln[(3 + 2√2)/√7] = 2ln[(√8 + 1)/(√7 + 0)]

Matching: a = 8, b = 7, c = 0

∴ a + b + c = 15

Question:

Step-by-Step Solution

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